Does the Skorokhod Banach space $D[0,1]$ (cadlag functions equipped with the uniform norm) admit a smooth partition of unity? I found Johanis - Smooth partitions of unity on Banach spaces, which provides several classes of Banach spaces with this property, but don't see how to relate the given criteria to the Skorokhod space.
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3$\begingroup$ The Skorohod space is not normed and thus not Banach. $\endgroup$– Iosif PinelisCommented Aug 4, 2020 at 14:07
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$\begingroup$ @IosifPinelis Maybe it is a matter of notation: What I'm considering are the cadlag functions together with the topology given by the uniform norm. This is a non-separable Banach space (and as far as I know, non-separability was the main reason why Skorokhod introduced the four more common J and M topologies) $\endgroup$– r_faszanatasCommented Aug 4, 2020 at 16:18
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1$\begingroup$ With the supremum norm topology the Skorohod space is linearly homeomorphic to $C([0,1])\oplus\ell^1([0,1])$ where neither or the factors has smooth bump functions by old results. Hence by reductio ad absurdum the answer is no, provided that the question is interpreted in the sense that every open cover has a subordinated smooth partition of unity.. $\endgroup$– TaQCommented Aug 6, 2020 at 5:49
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$\begingroup$ @TaQ thanks for this point of view, apparently there is something I don't understand, as in the reference I quoted it is explicitly stated that $C([0,1])$ does admit a partition of unity. I will read some more and then comment further. $\endgroup$– r_faszanatasCommented Aug 6, 2020 at 9:43
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$\begingroup$ @TaQ I think that I understand now: A necessary condition for $C(K)$ to admit smooth partitions of unity is that $K$ be scattered (has no nonempty dense-in-itself subsets), which rules out intervals of the real line (so my previous comment was wrong). Would you mind extending your comment to an answer, so that I can accept it? In particular, could you provide some more details on the linear homeomorphism (the definition is clear, but continuity requires some work, doesn't it)? $\endgroup$– r_faszanatasCommented Aug 6, 2020 at 10:43
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1 Answer
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The Skorohod space $D$ has $C([0,1])$ as a Banach subspace that does not have smooth bump functions by results of Bonic and Frampton from 1965. If $D$ had smooth partitions of unity, it would posses smooth bump functions, hence also $C([0,1])$ which is false. So the answer is no.
See Section 14 (pp. 152−158, in particular 14.11(1)) in Kriegl and Michor's The Convenient Setting of Global Analysis (AMS 1997) for an account on bump functions.